Theorem ltexprlemloc 7922

Description: Our constructed difference is located. Lemma for ltexpri 7928. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemloc	|- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r

Proof of Theorem ltexprlemloc

Dummy variables z w f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7724	. . . . . 6 |- ( q <Q r -> E. w e. Q. ( q +Q w ) = r )
2	1	adantl 277	. . . . 5 |- ( ( A <P B /\ q <Q r ) -> E. w e. Q. ( q +Q w ) = r )
3		ltrelpr 7820	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
4	3	brel 4802	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
5	4	simpld 112	. . . . . . . 8 |- ( A <P B -> A e. P. )
6		prop 7790	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7818	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
8	6, 7	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
9	5, 8	sylan 283	. . . . . . 7 |- ( ( A <P B /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
10	9	ad2ant2r 509	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
11	4	simprd 114	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
12	11	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> B e. P. )
13	12	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> B e. P. )
14		ltanqg 7715	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
15	14	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16		elprnqu 7797	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
17	6, 16	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
18	5, 17	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
19	18	adantlr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ q <Q r ) /\ y e. ( 2nd ` A ) ) -> y e. Q. )
20	19	ad2ant2rl 511	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> y e. Q. )
21		elprnql 7796	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
22	6, 21	sylan 283	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
23	5, 22	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
24	23	adantlr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( A <P B /\ q <Q r ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	24	ad2ant2r 509	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> z e. Q. )
26		simplrl 537	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> w e. Q. )
27		addclnq 7690	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) e. Q. )
28	25, 26, 27	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q w ) e. Q. )
29		ltrelnq 7680	. . . . . . . . . . . . . . . . . . 19 |- <Q C_ ( Q. X. Q. )
30	29	brel 4802	. . . . . . . . . . . . . . . . . 18 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
31	30	simpld 112	. . . . . . . . . . . . . . . . 17 |- ( q <Q r -> q e. Q. )
32	31	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ q <Q r ) -> q e. Q. )
33	32	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> q e. Q. )
34		addcomnqg 7696	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
35	34	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
36	15, 20, 28, 33, 35	caovord2d 6224	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( ( z +Q w ) +Q q ) ) )
37		addassnqg 7697	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ w e. Q. /\ q e. Q. ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
38	25, 26, 33, 37	syl3anc 1274	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
39		addcomnqg 7696	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Q. /\ q e. Q. ) -> ( w +Q q ) = ( q +Q w ) )
40	26, 33, 39	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( w +Q q ) = ( q +Q w ) )
41	40	oveq2d 6066	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( w +Q q ) ) = ( z +Q ( q +Q w ) ) )
42		simplrr 538	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( q +Q w ) = r )
43	42	oveq2d 6066	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( q +Q w ) ) = ( z +Q r ) )
44	38, 41, 43	3eqtrd 2269	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q r ) )
45	44	breq2d 4121	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( y +Q q ) <Q ( ( z +Q w ) +Q q ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
46	36, 45	bitrd 188	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
47	46	biimpa 296	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( y +Q q ) <Q ( z +Q r ) )
48		prop 7790	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
49		prloc 7806	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
50	48, 49	sylan 283	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
51	13, 47, 50	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
52	51	ex 115	. . . . . . . . . 10 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
53	52	anassrs 400	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) /\ y e. ( 2nd ` A ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
54	53	reximdva 2644	. . . . . . . 8 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) -> ( E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
55	54	reximdva 2644	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
56		prml 7792	. . . . . . . . . . . 12 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 1st ` A ) )
57		rexex 2588	. . . . . . . . . . . 12 |- ( E. z e. Q. z e. ( 1st ` A ) -> E. z z e. ( 1st ` A ) )
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 |- ( A e. P. -> E. z z e. ( 1st ` A ) )
59		r19.45mv 3603	. . . . . . . . . . 11 |- ( E. z z e. ( 1st ` A ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
60	5, 58, 59	3syl 17	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
61	60	adantr 276	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
62		prmu 7793	. . . . . . . . . . . . 13 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
63		rexex 2588	. . . . . . . . . . . . 13 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. x x e. ( 2nd ` A ) )
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 |- ( A e. P. -> E. x x e. ( 2nd ` A ) )
65		r19.43 2701	. . . . . . . . . . . . 13 |- ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
66		r19.9rmv 3601	. . . . . . . . . . . . . 14 |- ( E. x x e. ( 2nd ` A ) -> ( ( z +Q r ) e. ( 2nd ` B ) <-> E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
67	66	orbi2d 798	. . . . . . . . . . . . 13 |- ( E. x x e. ( 2nd ` A ) -> ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
68	65, 67	bitr4id 199	. . . . . . . . . . . 12 |- ( E. x x e. ( 2nd ` A ) -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 |- ( A <P B -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
70	69	rexbidv 2543	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
71	70	adantr 276	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
72		ltexprlem.1	. . . . . . . . . . . . . 14 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
73	72	ltexprlemell 7913	. . . . . . . . . . . . 13 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
74	72	ltexprlemelu 7914	. . . . . . . . . . . . . 14 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
75		eleq1 2295	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
76		oveq1 6057	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q r ) = ( z +Q r ) )
77	76	eleq1d 2301	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q r ) e. ( 2nd ` B ) <-> ( z +Q r ) e. ( 2nd ` B ) ) )
78	75, 77	anbi12d 473	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
79	78	cbvexv 1968	. . . . . . . . . . . . . . 15 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
80	79	anbi2i 457	. . . . . . . . . . . . . 14 |- ( ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
81	74, 80	bitri 184	. . . . . . . . . . . . 13 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
82	73, 81	orbi12i 772	. . . . . . . . . . . 12 |- ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
83		ibar 301	. . . . . . . . . . . . . . 15 |- ( q e. Q. -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
84	83	adantr 276	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
85		ibar 301	. . . . . . . . . . . . . . 15 |- ( r e. Q. -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
86	85	adantl 277	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
87	84, 86	orbi12d 801	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
88	30, 87	syl 14	. . . . . . . . . . . 12 |- ( q <Q r -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
89	82, 88	bitr4id 199	. . . . . . . . . . 11 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
90		df-rex 2526	. . . . . . . . . . . 12 |- ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
91		df-rex 2526	. . . . . . . . . . . 12 |- ( E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
92	90, 91	orbi12i 772	. . . . . . . . . . 11 |- ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
93	89, 92	bitr4di 198	. . . . . . . . . 10 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
94	93	adantl 277	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
95	61, 71, 94	3bitr4rd 221	. . . . . . . 8 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
96	95	adantr 276	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
97	55, 96	sylibrd 169	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
98	10, 97	mpd 13	. . . . 5 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
99	2, 98	rexlimddv 2665	. . . 4 |- ( ( A <P B /\ q <Q r ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
100	99	ex 115	. . 3 |- ( A <P B -> ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
101	100	ralrimivw 2616	. 2 |- ( A <P B -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
102	101	ralrimivw 2616	1 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   +Q cplq 7597   <Q cltq 7600   P.cnp 7606   <P cltp 7610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  ltexprlempr  7923